﻿14 
  T3r. 
  J. 
  W. 
  Nicliolson 
  on 
  the 
  Relation 
  of 
  

  

  of 
  magnitude 
  (n'-z)/2. 
  It 
  follows 
  that 
  all 
  ihe 
  higher 
  Bessel 
  

   functions 
  whose 
  order 
  and 
  argument 
  do 
  not 
  differ 
  widely 
  may 
  

   be 
  asymptotically 
  expressed 
  by 
  means 
  of 
  similar 
  functions 
  ot* 
  

   order 
  |. 
  

  

  In 
  the 
  first 
  place, 
  z 
  may 
  be 
  supposed 
  greater 
  than 
  n. 
  

   Thus 
  p 
  is 
  negative, 
  and 
  

  

  where 
  a= 
  — 
  p, 
  and 
  the 
  functions 
  J 
  have 
  argument 
  

  

  2o- 
  

  

  Moreover, 
  

  

  and 
  therefore 
  on 
  reduction 
  

  

  W=3V(|-»){ji 
  + 
  J^i} 
  (31) 
  

  

  J_,,(^) 
  = 
  !^W^('^.^-^j| 
  Jicos 
  (/^7^-j-i7^)^- 
  J_i 
  cos 
  («7r--j7r) 
  j 
  . 
  (32) 
  

  

  Similarly, 
  or 
  direct 
  from 
  the 
  definition, 
  in 
  the 
  case 
  of 
  n 
  

   integral, 
  

  

  Y„(^-)='r^(4-^-»){j-i-Jjl- 
  • 
  • 
  (33) 
  

  

  Another 
  function 
  of 
  great 
  importance 
  in 
  physical 
  theory 
  

   may 
  be 
  defined 
  by 
  

  

  K»(-)=2^^.^'"'"MJ-W-«'""J«(^)}- 
  • 
  (34) 
  

  

  Many 
  other 
  definitions 
  have 
  been 
  proposed, 
  but 
  this, 
  which 
  

   is 
  due 
  to 
  Macdonald*, 
  is 
  the 
  most 
  convenient 
  for 
  practical 
  

   applications. 
  

  

  When 
  n 
  is 
  an 
  integer, 
  

  

  = 
  -^(Y.(.^)+t7rJ,(c)y"2 
  \ 
  (35) 
  

  

  evaluating 
  the 
  undetermined 
  form. 
  

  

  Kn{z) 
  is 
  the 
  type 
  of 
  function 
  occurring 
  in 
  all 
  hai'monic 
  

  

  * 
  Proc. 
  Loncl. 
  31atb. 
  Soc. 
  vol. 
  32. 
  

  

  